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Let $a$ and $b$ be complex numbers. If $a + b = 1$ and $a^2 + b^2 = 2,$ then what is $a^3 + b^3?$

 Jun 9, 2024

Best Answer 

 #1
avatar+88 
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Let's first solve for ab:

\( (a + b)^2 = a^2 + b^2 + 2ab\)

\(1^2 = 2 + 2ab\)

\( -1 = 2ab\)

\(ab = -\frac{1}{2}\)

Now, we can solve for a^3 + b^3:

\((a^2 + b^2)(a + b) = a^3 + b^3 + a^2b + b^2a\)

\((2)(1) = a^3 + b^3 + ab(a) + ab(b)\)

\(2 = a^3 + b^3 + ab(a + b)\)

\(2 = a^3 + b^3 + -\frac{1}{2}(1)\)

\(a^3 + b^3 = 2 + \frac{1}{2}\)

\(\mathbf{a^3 + b^3 = 5/2}\)

 Jun 9, 2024
 #1
avatar+88 
+1
Best Answer

Let's first solve for ab:

\( (a + b)^2 = a^2 + b^2 + 2ab\)

\(1^2 = 2 + 2ab\)

\( -1 = 2ab\)

\(ab = -\frac{1}{2}\)

Now, we can solve for a^3 + b^3:

\((a^2 + b^2)(a + b) = a^3 + b^3 + a^2b + b^2a\)

\((2)(1) = a^3 + b^3 + ab(a) + ab(b)\)

\(2 = a^3 + b^3 + ab(a + b)\)

\(2 = a^3 + b^3 + -\frac{1}{2}(1)\)

\(a^3 + b^3 = 2 + \frac{1}{2}\)

\(\mathbf{a^3 + b^3 = 5/2}\)

Maxematics Jun 9, 2024

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